Method of fabricating curved surfaces

ABSTRACT

Accurate and inexpensive fabrication of mathematically derived curved surfaces is achieved by deflecting a supported beam, preferably a cantilever beam, with a force applied to the free end of the beam. Deflection of the beam at any point along its length is determined, in part, by the moment of inertia of the beam cross-section at that point; thus by varying the beam crosssection along the beam length, the deflected beam is made to conform to the desired curve. The resulting accurately attained curved surface has primary utilization as a reflector of light, sound, microwave or other energy beams.

United States Patent [191 Weiser [451 Oct. 2, 1973 [52] US. Cl. 72/388, 343/840 [51] Int. Cl B2111 11/08 [58] Field of Search 72/388; 264/339;

[56] References Cited UNITED STATES PATENTS ll/l970 Berks et al 343/840 OTHER PUBLICATIONS Katz, Simple Parabolic Antenna Design, CQ August 1966, pp. 10-13, copy in 343/840 ENEQGY Knadle, Jr., A Twelve-Foot Stressed Parabolic Dish," QST, August 1972, pp. 16-22, copy in 343/840 Primary ExaminerLowell A. Larson Attorney-Rose and Edell 57 ABSTRACT Accurate and inexpensive fabrication of mathematically derived curved surfaces is achieved by deflecting a supported beam, preferably a cantilever beam, with a force applied to the free end of the beam. Deflection of the beam at any point along its length is determined,

in part, by the moment of inertia of the beam crosssection at that point; thus by varying the beam crosssection along the beam length, the deflected beam is made to conform to the desired curve. The resulting accurately attained curved surface has primary utilization as a reflector of light, sound, microwave or other energy beams.

10 Claims, 9 Drawing Figures BEAM 1 METHOD OF FABRICATING CURVED SURFACES BACKGROUND OF THE INVENTION The present invention relates to a method for accurately attaining specific curved surfaces. More particularly, the invention concerns the fabrication of reflectors by a method which is simpler and less expensive than has heretofore been possible.

One prior art method of fabricating curved reflectors, particularly optical reflectors, is by optically grinding and polishing a glass blank. Such a technique is quite accurate but quite costly and time consuming.

Another prior art method of fabricating a curved reflector is by utilizing multiple spaced screw adjustments to effect a piecemeal contouring of the surface of a thin reflecting member. This discrete point adjustment approach is relatively inexpensive but lacks sufficient accuracy for many, if not most, applications.

It is therefore an object of the present invention to fabricate curved reflective surfaces by a method which is accurate on the one hand yet inexpensive and fast on the other hand.

It is an object of the present invention to provide a simple yet accurate method of fabricating mathematically derived curved surfaces.

It is important to note that the method described herein is applicable to fabricating curved reflectors of substantially any type of beamed energy, such as light, sound, microwaves, etc.

SUMMARY OF THE INVENTION According to the principles of the present invention, a body to be formed into a curved reflector is suspended, for example, as a cantilever beam. The free end of the beam is loaded with a known force in order to deflect the beam. The amount of deflection at each point along the length of the beam is determined, in part, by the moment of inertia of the beam crosssection at that point. Thus by varying the beam crosssection along the beam length, the deflected beam is made to conform to a predetermined curve.

The suspended beam need not be cantilevered; rather it may be supported at its ends or at other locations and suitably deflected. The important aspect of this invention resides not so much in the manner of beam support as in the concept of varying the beam cross-section as necessary to attain the desired curve.

BRIEF DESCRIPTION OF THE DRAWINGS The above and still further objects, features and advantages of the present invention will become apparent upon consideration of the following detailed description of specific embodiments thereof, especially when taken in conjunction with the accompanying drawings, wherein:

FIG. 1 is a plan view in elevation of a cantilever beam prior to deflection according to the method of the present invention;

FIG. 2 is a view taken along lines 2-2 in FIG. 1;

FIG. 3 is a view taken along lines 3-3 in FIG. 1;

FIG. 4 is a plan view of the beam of FIG. 1 after deflection;

FIG. 5 is a plan view in elevation of a cantilever beam of different cross-sectional configuration from that of the beam of FIG. 1;

FIG. 6 is a view taken along lines 6-6 of FIG. 5;

FIG. 7 is a view taken along lines 77 of FIG. 5;

FIG. 8 is a diagrammatic illustration of the reflective properties of a beam deflected according to the method of the present invention; and

FIG. 9 is a diagrammatic illustration of the reflective properties of a modified beam deflected according to the method of the present invention.

DESCRIPTION OF PREFERRED EMBODIMENTS Referring specifically to FIG. 1 of the accompanying drawings, a beam 10 of length L is suspended cantilever fashion from support 11. The underside 12 of beam 10 is assumed to have reflective properties for the medium to be reflected; that is, if an optical reflector is to be fabricated, underside 12 is assumed to be highly reflective of light or at least capable of being polished to attain a high degree of reflectivity. As illustrated in FIGS. 2 and 3, the cross-section of beam 10 is rectangular and has a width b and a height h. The rectangular crosssection is taken as being illustrative only since it simplifies much of the analysis which follows. It is to be understood that the cross-section of the beam can assume a variety of configurations, such as triangular, rhombic, trapezoidal, etc., so long as underside 12 presents a reflective surface suitable for the intended utilization of the finished product.

It will be noted from FIGS. 2 and 3 that the width b of beam 10 is smaller proximate the free end of the beam than at the fixed end of the beam. In fact, for the particular reflector configuration considered below, the dimension b decreases linearly with distance from support 11. The linear variation of b results in corresponding linear changes in the moment of inertia of the beam cross-section along the beam length. In the manner described below, this cross-section variation permits the beam to be deflected to form a parabolic curve.

Assume a load in the form of force P to be applied at the free end of beam 10, acting to deflect the beam downwardly as illustrated in FIG. 4. The downward deflection y of the beam as a function of the displacement at from support 1 1 is described mathematically by equation (I) as follows (Reference: Elements of Strength of Materials, Timoshenko and MacCullough, D. Van Nostrand Co., 1949, p. 167):

y P/EI (1 /6 Lx /2) where E is Young's Modulus (modulus of elasticity) for the material of beam 10 and I is the moment of inertia of the cross-section of beam 10 at x. For a rectangular cross-section, the moment of inertia I is represented by equation (2) as follows (Reference: Timoshenko et al., p. 392):

I bh ll 2 For a beam having constant cross-section throughout its length, the moment of inertia I of the cross-section is also constant. Such a beam, when deflected, assumes a configuration which is not readily useful for energy reflection applications. I have found, however, that by varying the cross-section as necessary, the deflected beam can be forced to conform to a wide variety of use ful reflector configurations. For example, assume it is desired to obtain a parabolic reflector having a reflecting surface represented by the expression:

Substituting for y in expression (1 results in the following expression:

Simplifying expression (4) and solving for I as a function of 1: yields the following:

I P/6E (3L x).

It is noted that I is a linear function of 1:, assuming force P, length L, and modulus E to be constant. Substituting for I from expression (2) and simplifying yeiid the following Thus, by reducing dimension 1) of the beam as a linear function of displacement x from support 11, it is possible to obtain a parabolic surface 12 of the deflected beam. The cross-section views of FIGS. 2 and 3 indicate just such a linear reduction of b.

Of course, dimension h, rather than b, may be varied to obtain a parabolic surface. Under such circumstances expression (6) is properly written as follows:

h 2P/bE (3L x) The relationship between h and x is not linear; consequently the resulting beam configuration would not be as easy to achieve as in the case where b is varied with 1:. Likewise, both b and h can be varied with x to attain the desired curve. An example of this is illustrated in FIGS. 5, 6 and 7 in which a beam 20 has both its b and h dimensions decreasing with increasing values of 2:. To assume a resulting parabolic surface for such a beam, the following relationship must be adhered to:

bh ZP/E (3L x).

The description set forth above pertains to parabolic reflectors; however reflectors configured as other mathematically represented curves can be fabricated by the same method. For example, an ellipse, a hyperbole, a circle, or portions thereof, as well as other mathemtical functions, may be utilized as reflectors fabricated by the method of this invention. In each case it is first desirable to provide a mathematical expression for the desired reflector curve in terms of y =f(x). This expression is then substituted for y in expression (1) and the resulting expression is solved for I in terms of x. The formula for I (which depends upon the general configuration-of the beam cross-section, i.e. rectangular, triangular, etc.) is then obtained (either from a reference table or by mathematical derivation) and substituted into the expression for I. This results in an expression for one or more cross-section dimensions in terms of x. This latter expression is then utilized to configure the beam. It is assumed, of course, that values for P and L are chosen in advance and that the value of E is known.

The deflected beam, as mentioned above, either has a reflective surface 12 to begin with or has a surface 12 capable of being readily processed to be reflective. The type of processing, of course, depends upon the medium being reflected, whether it be light, sound, microwave energy, etc. An example of a finished reflector of the parabolic type is illustrated in FIG. 8. An energy beam, for example a radar beam, is directed toward reflective surface 12 of beam 10 from the focal point 15 of the parabolic formed by beam 10. The energy beam may be derived from an energy (e.g. microwave energy) source located at focal point 15; alternatively, the energy source may be displaced from the focal point but arranged to direct its beam toward a reflector 16 located at the focal point and arranged to direct the beam toward reflective surface 12. In either case, the reflected energy rays from any point on surface 12 are parallel, a well-known characteristic of parabolic reflectors which are fed energy from their focal points.

A modification of the method described above is illustrated in FIG. 9 and comprises uniformly bonding a thin flexible reflective sheet 14 to the undersurface 12 of the beam 10 before deflection. The sheet l4 conforms quite well to curves of surface 12, particularly if the h dimension of sheet 14 is small compared to the h dimension of beam 10. FIG. 9 also illustrates an optical system wherein beam 10 has been deflected to provide a parabolic optical reflector having a small cylindrical mirror 18 located at its focus point. A laser beam is directed toward and reflected by mirror 18 which has negligible effect on beam divergence. If the radius of mirror 18 is kept small relative to minimum radius of curvature of the parabolic reflector, parallel rays of light are reflected from the parabolic reflector.

It must again be stressed that although a cantilever beam is the simplest arrangement for employing the method of the present invention, the inventive concept includes substantially any mode of supporting the beam. The crucial aspect of the invention is the proper configuration of the beam cross-section along its length so as to produce the desired configuration upon deflection.

While I have described and illustrated specific embodiments of my invention, it will be clear that variations of the details of constructionwhich are specifically illustrated and described may be resorted to without departing from the true spirit and scope of the invention as defined in the appended claims.

I claim:

1. The method of fabricating a curved surface which conforms to a predetermined mathematical curve, said method comprising the steps of:

forming a member of specified material and length with a cross-section perpencidular to said length which varies along said length;

supporting said member at least at one point along its length; and

applying a known force to said member perpendicular to said length to deflect said member into the configuration of said predetermined mathematical curve;

wherein said step of forming includes varying said cross-section of said member such that the moment of inertia of said cross-section at each point along said length is equal to that required for the overall length of said member to conform to said predetermined mathematical curve upon application of said known force.

2. The method according to claim 1 further comprising the step of rendering one surface of said member reflective to beams of a specified energy medium.

3. The method according to claim 1 further comprising the step of polishing one surface of said member to render it optically reflective.

4. The method according to claim 1 wherein said step of supporting includes fixedly supporting one end of said member such that said member is suspended horizontally in cantilever fashion, said known force being applied at the free end of said member.

5. The method according to claim 4 wherein said step of forming includes configuring said cross-section such that said moment of inertia decreases linearly along said member as the distance from said one end increases.

6. The method of fabricating a curved surface to conform to a mathematical curve y =f (x), said method including the deflection of a member in one direction, y, by a force acting perpendicular to the direction, x, along the length of said member, said method comprising the steps of:

forming said member with a cross-section perpendicular to direction x which varies such that the moment of inertia I of said cross-section is represented as I =f2( supporting said member horizontally such that upon application of a known vertical force thereto each point along the length of said member is deflected as a function of the moment of inertia of said crosssection at that point and the location of that point along the length of said member, the expression for said deflection being represented by the expression y =fl( ),fa( and applying said known vertical force at a predetermined location along the length of said member at a level sufficient to deflect said member to a configuration represented by said mathematical curve =f1( wherein said step of forming includes configuring said cross-section such that the moment of inertia of said cross-section is determined by setting f (x) =fiu),fa( yfzo) =f1( f3( 7. The method according to claim 6 wherein said mathematical curve is a parabolic of the form y Ax B, where A and B are constants, wherein said step of supporting includes fixing one end of said member and suspending the other end freely in cantilever arrangement, wherein said known vertical force is applied at said free end of said member, wherein said expression for deflection y =f (I),f (x) is equal to y P/EI (r /6 L/2)'where P is said known vertical force, E is the modulus of elasticity of the material of said member and L is the length of said member, and wherein the moment of inertia I f (x) is represented by the expression P/6E (3L x).

8. The method according to claim 7 wherein said cross-section of said member is rectangular in configuration, decreasing in size from said fixed end to said free end of said member.

9. The method according to claim 6 further comprising the step of rendering the undersurface of said member reflective of beams of a predetermined energy medium.

10. The method according to claim 9 wherein said energy medium is light energy. 

1. The method of fabricating a curved surface which conforms to a predetermined mathematical curve, said method comprising the steps of: forming a member of specified material and length with a crosssection perpencidular to said length which varies along said length; supporting said member at least at one point along its length; and applying a known force to said member perpendicular to said length to deflect said member into the configuration of said predetermined mathematical curve; wherein said step of forming includes varying said cross-section of said member such that the moment of inertia of said crosssection at each point along said length is equal to that required for the overall length of said member to conform to said predetermined mathematical curve upon application of said known force.
 2. The method according to claim 1 further comprising the step of rendering one surface of said member reflective to beams of a specified energy medium.
 3. The method according to claim 1 further comprising the step of polishing one surface of said member to render it optically reflective.
 4. The method according to claim 1 wherein said step of supporting includes fixedly supporting one end of said member such that said member is suspended horizontally in cantilever fashion, said known force being applied at the free end of said member.
 5. The method according to claim 4 wherein said step of forming includes configuring said cross-section such that said moment of inertia decreases linearly along said member as the distance from said one end increases.
 6. The method of fabricating a curved surface to conform to a mathematical curve y f1(x), said method including the deflection of a member in one direction, y, by a force acting perpendicular to the direction, x, along the length of said member, said method comprising the steps of: forming said member with a cross-section perpendicular to direction x which varies such that the moment of inertia I of said cross-section is represented as I f2(x); supporting said member horizontally such that upon application of a known vertical force thereto each point along the length of said member is deflected as a function of the moment of inertia of said cross-section at that point and the location of that point along the length of said member, the expression for said deflection being represented by the expression y f4(I), f3(x); and applying said known vertical force at a predetermined location along the length of said member at a level sufficient to deFlect said member to a configuration represented by said mathematical curve y f1(x); wherein said step of forming includes configuring said cross-section such that the moment of inertia of said cross-section is determined by setting f1(x) f4(I), f3(x), whereby f2(x) f1(x), f3(x).
 7. The method according to claim 6 wherein said mathematical curve is a parabolic of the form y Ax2 + B, where A and B are constants, wherein said step of supporting includes fixing one end of said member and suspending the other end freely in cantilever arrangement, wherein said known vertical force is applied at said free end of said member, wherein said expression for deflection y f4(I), f3(x) is equal to y P/EI (x3/6 - L2/2) where P is said known vertical force, E is the modulus of elasticity of the material of said member and L is the length of said member, and wherein the moment of inertia I f2(x) is represented by the expression P/6E (3L - x).
 8. The method according to claim 7 wherein said cross-section of said member is rectangular in configuration, decreasing in size from said fixed end to said free end of said member.
 9. The method according to claim 6 further comprising the step of rendering the undersurface of said member reflective of beams of a predetermined energy medium.
 10. The method according to claim 9 wherein said energy medium is light energy. 